Collective Excitation and Quantum Depletion of a Bose–Einstein Condensate in a Periodic Array of Quantum Wells

With the help of a set of exact closed-form solutions to the stationary Gross–Pitaevskii (GP) equation, we calculate the collective excitation and quantum depletion of a weakly interacting Bose gas in the presence of a periodic array of quantum wells. The excitation spectrum (Bogoliubov spectrum) is obtained from the solution of the linearized time-dependent GP equation, which develops energy bands ?ω_j( p) periodic in quasi-momentum space. Moreover, we calculate the excitation strengths Z_j( p) relative to the density operator and then the dynamic structure factor S( p,ω). Accordingly, the analytical expressions of quantum depletion of the system are obtained. We find that the quantum depletion is enhanced when the interatomic interactions become larger and the potential is sufficiently deep. The conditions for the possible experimental realization of our scenario are also proposed.